Temporal graphs are graphs whose edge relation changes in discrete time steps. The Temporal Exploration Problem (TEXP) asks for a temporal walk that visits all vertices. We show that every connected temporal graph with $n$ vertices can be explored in $O(qn\log^3(n))$ time steps, where $q$ is the number of vertex color classes induced by the 2-dimensional Weisfeiler-Leman (2-WL) coloring. For small $q$, our bound substantially improves on the known worst-case bound of $Θ(n^2)$ in general connected temporal graphs. A corresponding walk can be constructed in deterministic polynomial time. The upper bound is complemented by lower bounds of $Ω(n\log(n))$ when $q$ is constant and $Ω(qn)$ when $1 \leq q \leq n$. We also apply our techniques to the Temporal Rendezvous Problem (TRP), in which two agents that do not communicate have to meet on a vertex of the temporal graph. Our exploration algorithm yields that, with only polynomial-time computational power, the agents can meet in $O(n\log^3(n))$ time steps, leaving a gap of only $\log^2(n)$ to the known lower bound. Dogeas, Erlebach, Kammer, Meintrup, and Moses Jr. [ICALP'24] have shown an $O(rn^{1+ε})$ bound for TEXP, for every fixed $ε> 0$, where $r$ is the number of orbits, and an $O(n^{1+ε})$ bound for TRP. Both results require superpolynomial computation time. Our contribution improves upon these results in several ways. First, the dependence on $n$ in the upper bounds on the number of time steps is decreased from $n^{1+ε}$ to $n\log^3(n)$ for both TEXP and TRP. Second, for TEXP, we bound the number of time steps linearly by the parameter q, which is efficiently computable and upper bounded by $r$. Third, we provide novel deterministic polynomial-time algorithms for both TEXP and TRP.
🌊
